Definition

Explicitly, an AbAb-enriched category is a categoryCC such that for all objects a,ba,b the hom-setHomC(a,b)Hom_C(a,b) is equipped with the structure of an abelian group; and such that for all triples a,b,ca,b,c of objects the composition operation ∘a,b,c:HomC(a,b)×HomC(b,c)→HomC(a,c)
\circ_{a,b,c} : Hom_C(a,b) \times Hom_C(b,c) \to Hom_C(a,c)
is bilinear. A ringoid is small AbAb-enriched category.

Remarks

There is a canonical forgetful functor Ab→Set*Ab \to Set_* from abelian groups to pointed sets, which sends each group to its underlying set with point being the neutral element. Using this functor, every AbAb-enriched category CC is in particular also a category that is enriched over pointed sets (that is, a category with zero morphisms). This is sufficient for there to be a notion of kernel and cokernel in CC.

Finite products are absolute

One of the remarkable facts about AbAb-enriched categories is that finite products (and coproducts) are absolute limits. This implies that finite products coincide with finite coproducts, and are preserved by anyAbAb-enriched functor.

Zero objects

In an AbAb-enriched category CC, any initial object is also a terminal object, hence a zero object, and dually. An object a∈Ca\in C is a zero object just when its identity 1a1_a is equal to the zero morphism 0:a→a0:a\to a (that is, the identity element of the abelian group homC(a,a)\hom_C(a,a)). Expressed in this way, it is easy to see that any AbAb-enriched functor preserves zero objects.

Biproducts

For c1,c2∈Cc_1, c_2 \in C two objects in an AbAb-enriched category CC, theproductc1×c2c_1 \times c_2 coincides with thecoproductc1⊔c2c_1 \sqcup c_2 when either exists. More precisely, when both exist, the canonical morphism

which exists whenever c1⊔c2c_1\sqcup c_2 and c1×c2c_1\times c_2 do, is an isomorphism. This object is called a biproduct or (sometimes) a direct sum and is generally denoted

c1⊕c2.
c_1 \oplus c_2.

It can be characterized diagrammatically as an object c1⊕c2c_1\oplus c_2 equipped with morphisms qi:ci→c1⊕c2q_i:c_i\to c_1\oplus c_2 and pi:c1⊕c2→cip_i:c_1\oplus c_2 \to c_i such that piqj=δijp_i q_j = \delta_{i j} and q1p1+q2p2=1c1⊕c2q_1 p_1 + q_2 p_2 = 1_{c_1\oplus c_2}. Expressed in this form, it is clear that any AbAb-enriched functor preserves biproducts.

As a generalisation of rings

When using the term ‘ringoid’, one often assumes a ringoid to be small.

Examples

For any small AbAb-enriched category RR, the enriched presheaf category[Rop,Ab][R^{op},Ab] is, of course, AbAb-enriched. If RR is a ring, as above, then [Rop,Ab][R^{op},Ab] is the category of RR-modules.